An option to buy a stock is priced at $$200.$ If the stock closes above $30$ on May $15,$ the option will be worth $$600.$ If it closes below $20,$ the option will be worth nothing, and if it closes between $20$ and $30($inclusively$),$ the option will be worth $$200.$ A trader thinks there is a $40\%$ chance that the stock will close in the $20-30$ range, a $50\%$ chance that it will close above $30,$ and a $10\%$ chance that it will fall below $20$ on May $15.$ Complete parts $($a$)$ through $($c$).$

a$)$ How much does she expect to gain?

$ $($Round to the nearest dollar as needed.$)$

b$)$ What is the standard deviation of her gain?

$ $($Round to the nearest dollar as needed.$)$

c$)$ Should she buy the stock option? Discuss the pros and cons in terms of your answers to $($a$)$ and $($b$).$

The expected gain is $$ which means the trader can expect The standard deviation is

$$ than the expected gain. That means a net loss is $$ The trader should balance the risk of a net loss against her tolerance for loss to make the decision.